Motion and distinguishing number two

A group A acting faithfully on a finite set X is said to have distinguishing number two if there is a proper subset Y whose (setwise) stabilizer is trivial. The motion of A acting on X is defined as the largest integer k such that all non-trivial elements of A move at least k elements of X. The Motion Lemma of Russell and Sundaram states that if the motion is at least 2 log(2) vertical bar A vertical bar, then the action has distinguishing number two. When X is a vector space, group, or map, the Motion Lemma and elementary estimates of the motion together show that in all but finitely many cases, the action of Aut (X) on X has distinguishing number two. A new lower bound for the motion of any transitive action gives similar results for transitive actions with restricted point-stabilizers. As an instance of what can happen with intransitive actions, it is shown that if X is a set of points on a closed surface of genus g, and vertical bar X vertical bar is sufficiently large with respect to g, then any action on X by a finite group of surface homeomorphisms has distinguishing number two.